E. Abu-Nada
e-mail: [email protected]
A. Al-Sarkhi
B. Akash
I. Al-Hinti
Department of Mechanical Engineering,
Hashemite University,
Zarqa, 13115, Jordan
Heat Transfer and Fluid Flow
Characteristics of Separated
Flows Encountered in a
Backward-Facing Step Under the
Effect of Suction and Blowing
Numerical investigation of heat transfer and fluid flow over a backward-facing step
(BFS), under the effect of suction and blowing, is presented. Here, suction/blowing is
implemented on the bottom wall (adjacent to the step). The finite volume technique is
used. The distribution of the modified coefficient of friction and Nusselt number at the top
and bottom walls of the BFS are obtained. On the bottom wall, and inside the primary
recirculation bubble, suction increases the modified coefficient of friction and blowing
reduces it. However, after the point of reattachment, mass augmentation causes an increase in the modified coefficient of friction and mass reduction causes a decrease in
modified coefficient of friction. On the top wall, suction decreases the modified coefficient
of friction and blowing increases it. Local Nusselt number on the bottom wall is increased
by suction and is decreased by blowing, and the contrary occurs on the top wall. The
maximum local Nusselt number on the bottom wall coincides with the point of reattachment. High values of average Nusselt number on the bottom wall are identified at high
Reynolds numbers and high suction bleed rates. However, the low values correspond to
high blowing rates. The reattachment length and the length of the top secondary recirculation bubble are computed under the effect of suction and blowing. The reattachment
length is increased by increasing blowing bleed rate and is decreased by increasing
suction bleed rate. The spots of high Nusselt number, and low coefficient of friction, are
identified by using contour maps. 关DOI: 10.1115/1.2759973兴
Keywords: separated flows, heat transfer, backward-facing step, blowing, suction, porous wall
Introduction
Heat transfer and fluid flow in separated flows are frequently
encountered in various engineering applications. Some examples
include microelectronic circuit boards, combustors, heat exchangers, axial and centrifugal compressor blades, and gas turbines
blades. The flow over a backward-facing step 共BFS兲 has the most
basic features of separated flows, such as separation, reattachment, recirculation, and development of shear layers. It is well
known that heat transfer and fluid flow characteristics experience
large variation within separated regions. Thus, it is very essential
to understand the mechanisms of heat transfer in such regions in
order to enhance heat and fluid flow.
Most of the published work on BFS has been extensively investigated for impermeable walls. For example, Armaly et al. 关1兴
studied laminar, transition, and turbulent isothermal flows over a
BFS, experimentally. In their experiments, the expansion ratio
was close to 2 and the downstream aspect ratio close to 18. Also,
numerical studies in the laminar regime for isothermal flows were
conducted by Armaly et al. 关1兴, Gartling 关2兴, and Kim and Moin
关3兴. Thangam and Knight 关4兴 studied the effect of step height on
the separated flow past a backward-facing step. On the other hand,
flow over a BFS with heat transfer was conducted 关5–17兴. AbuMulaweh 关18兴 conducted an extensive review of research on laminar mixed convection over a BFS.
Contributed by the Heat Transfer Division of ASME for publication in JOURNAL OF
HEAT TRANSFER. Manuscript received August 22, 2006; final manuscript received
February 1, 2007. Review conducted by Louis C. Burmeister.
Journal of Heat Transfer
Three-dimensional studies were also conducted for flow over a
BFS. A recent study by Williams and Baker 关19兴 focused on threedimensional numerical simulations of laminar flow over a step
with sidewalls, having the same expansion ratio and aspect ratio
of Armaly et al. 关1兴. The range of the Reynolds number studied
was 100⬍ Re⬍ 800. Williams and Baker 关19兴 reported that “the
interaction of a wall jet located at the step plane to the side walls
with the mainstream flow causes a penetration of threedimensional flow structure into flow near the mid plane of the
channel.” This penetration reduces the size of the upper secondary
zone compared to two-dimensional simulation, which leads to an
increase in reattachment length of the primary zone with Reynolds
number. Also, they reported that the observed threedimensionality is not caused by an inherent hydrodynamic instability of the two-dimensional base flow, but rather by the boundary conditions imposed by the sidewalls.
Moreover, Tylli et al. 关20兴 focused on three-dimensional effects
induced by the presence of sidewalls. They conducted particle
image velocimetry 共PIV兲 measurements accompanied with threedimensional numerical simulations, for an expansion ratio of 2
and a downstream aspect ratio of 20. They showed that for Re
⬍ 400 the sidewall effects do not affect the structure of the twodimensional flow at the channel midplane. Also, they illustrated
the existence of a wall-jet moving toward the channel midplane
that causes the three-dimensionality of the flow. The wall jet intensity increases with Reynolds number for laminar flow. Moreover, they studied the effect of sidewalls on the upper secondary
recirculation bubble and on the primary bubble and showed that,
Copyright © 2007 by ASME
NOVEMBER 2007, Vol. 129 / 1517
for laminar flow, these effects explain the discrepancies between
experimental results and two-dimensional numerical simulations.
Chiang and Sheu 关21兴 performed three-dimensional numerical
simulations for laminar flow over a BFS with the same geometry
dimensions used by Armaly et al. 关1兴. They reported that the flow
structure at the midplane, for Re= 800, is similar to twodimensional flow for channel width of up to 100 times of the step
height. Further three-dimensional investigations were also conducted by Chiang and Sheu 关22,23兴.
In terms of fundamental studies of the flow stability, Kaiktsis et
al. 关24兴 have investigated the effects of convective instabilities.
Barkley et al. 关25兴 have shown that in the absence of sidewalls the
transition to three-dimensional flow structures appears at higher
values of Reynolds number around 1000, via a steady bifurcation.
They found that the critical eigenmode responsible for threedimensionality is characterized by flat streamwise rolls located in
the primary recirculation bubble.
Studies on BFS with permeable walls, by means of suction and
blowing, have also been conducted. For example, Yang et al. 关26兴
studied the effect of mass bleed in separated reattaching turbulent
flow behind a BFS, experimentally. Batenko and Terekhov 关27兴
conducted two-dimensional simulation for unconfined flow past a
step and flow evolution in a confined channel. Kaiktsis and
Monkewitz 关28兴 studied the global destabilization of flow over a
BFS by means of simultaneous suction and blowing, using numerical simulations. The extensive interest and research conducted on BFSs, in the last two decades, with the complex physics
encountered in separated flow over the BFS has motivated the
present investigation. The objective of the present work is to perform a detailed study on the effect of suction and blowing on the
heat transfer and flow characteristics of steady flow over a BFS.
Governing Equations
The nondimensional continuity, momentum, and energy equations in Cartesian coordinates for steady flow are given as 关29兴
⳵u ⳵v
+
=0
⳵x ⳵y
u
u
共1兲
冉
冉
⳵u
⳵u
⳵ p 1 ⳵ 2u ⳵ 2u
+v =−
+
+
⳵x
⳵y
⳵ x Re ⳵ x2 ⳵ y 2
⳵v
⳵v
⳵ p 1 ⳵ 2v ⳵ 2v
+v =−
+
+
⳵x
⳵y
⳵ y Re ⳵ x2 ⳵ y 2
u
冉
⳵␪
⳵␪
1
⳵ 2␪ ⳵ 2␪
+v =
+
⳵x
⳵ y Re Pr ⳵ x2 ⳵ y 2
冊
冊
共2兲
共3兲
冊
共4兲
where
Re =
The basic flow configuration under study is shown in Fig. 1.
The expansion ratio 共ER= H / S兲 is set equal to 2.0. In the x direction, the physical domain is bounded by 0 ⬍ x ⬍ 共30H兲. Also, in
the y direction the physical domain is bounded by −共1 / 2兲 ⬍ y
⬍ 共1 / 2兲. The flow is assumed two-dimensional, steady, incompressible, and having constant fluid properties.
Part of the channel’s bottom wall, adjacent to the step, is considered permeable and constant uniform velocity is allowed to
bleed through it 共see Fig. 1兲. The shaded area represents the porous segment of the bottom wall. In order to study the effect of
mass bleed on the entire primary recirculation bubble on the lower
wall, the length of the porous segment must be at least equal to the
reattachment length. It is very well documented in literature that,
for Re⬍ 800, the reattachment length will not exceed a value of
10DH 关1–7兴. Thus, this length was selected for the present study.
The arrows that are shown on the porous segment represent the
direction of mass bleed through the wall. Inward direction represents blowing in the domain, and outward direction represents
suction flow out of the domain.
At the inlet, the flow is assumed hydrodynamically fully developed, where a dimensionless parabolic velocity distribution is
given as 关30兴
u共y兲 = 12共y − 2y 2兲
共5兲
A no-slip velocity boundary condition is applied at the top wall
of the channel and at the vertical wall of the step because these
two walls are assumed impermeable. However, only the x component of velocity is set equal to zero at the bottom wall, where
normal bleed velocity is imposed 共for x ⬍ 10DH兲. However, for
x ⬎ 10DH a no-slip boundary condition is also imposed on the
bottom wall.
A fully developed velocity is assumed at the channel outlet
⳵u ⳵v
=
=0
⳵x ⳵x
␷
␣
The dimensional y component of velocity at the porous segment is
* . A nondimensional bleed coefficient is defined as
set equal to v
w
关31兴:
The following dimensionless quantities are used:
*y
y=
,
DH
Problem Description
u mD H
␷
Pr =
*x
x=
,
DH
Fig. 1 Sketch of the problem geometry and boundary
conditions
*u
u= ,
um
*
v
v= ,
um
p=
*p
,
␳um2
and
T − Twc
␪=
Twh − Twc
where Twh is the constant temperature of the hot wall and Twc is
the constant temperature of the cold wall.
1518 / Vol. 129, NOVEMBER 2007
␴=
*
v
w
um
共6兲
共7兲
The values of bleed coefficient used in the present work are ± 共0,
0.00125, 0.0025, and 0.005兲, where positive values correspond to
blowing and negative values correspond to suction. The case of
zero value of bleed coefficient corresponds to impermeable wall.
Therefore, the nondimensional velocity at the porous segment is
equal to the bleed coefficient. This is expressed as
Transactions of the ASME
Fig. 2 „a… Computational mesh and „b… typical control volume
vw = ␴
共8兲
Note that a bleed coefficient of 0.005 corresponds to a blowing
volume flow rate equal to 10% of the volume flow rate at inflow.
The temperature at the inlet is assumed to be fully developed
and is given as 关32兴
␪ = 1 − 2y
共9兲
The temperature boundary condition at the outlet is given as
⳵␪
=0
⳵x
共10兲
The temperature is set as a constant value 共␪ = 1兲 on the step
vertical wall and on the bottom wall downstream of the step,
including the porous segment. The temperature at the top flat wall
is set to zero 共␪ = 0兲.
The total length of the computational domain is taken as 共L
= 30H兲 to ensure fully developed outlet boundary condition
关2,5,6,32兴.
冏 冏
⳵ 共u␾兲
⳵x
=
P
1
共− uEE␾EE + 4uE␾E − 3u P␾ P兲,
2⌬x
冏 冏
⳵ 共u␾兲
⳵x
=
P
1
共3u P␾ P − 4uW␾W + 3uWW␾WW兲,
2⌬x
for
u⬎0
共11兲
Journal of Heat Transfer
u⬍0
共12兲
Similar expressions could be written for the y direction. Figure 2
shows the computational mesh and the control volume with the
symbols used in Eqs. 共11兲 and 共12兲.
In the x direction, a fine grid is used in the regions near the
point of reattachment to resolve the steep velocity gradients while
a coarser grid is used downstream the point 关35兴. However, in the
y direction, a fine grid is used near the top, the bottom walls, and
directly at the step. Fine mesh in the x and y directions is generated by using an algebraic grid-stretching technique that results in
considerable savings in terms of the grid size and in computational time.
In the x direction, the grid-stretching method is implemented by
constructing uniformly distributed grid points in the x direction,
and then transforming these points into a nonuniform mesh. The
transformation is given as 关35,36兴
再
Numerical Implementation
Equations 共1兲–共4兲, with corresponding boundary conditions
共i.e., Eqs. 共5兲, 共6兲, and 共8兲–共10兲兲, are solved using the finite volume approach 关33,34兴. The SIMPLE algorithm is used as the computational algorithm 关33,34兴. The diffusion term in the momentum
and energy equations is approximated by second-order central difference which gives a stable solution. However, a second-order
upwind differencing scheme is adopted for the convective terms.
This scheme uses second-order extrapolation of two upwind
neighbors to determine any transport quantity ␾. The secondorder upwind term is written in the following general form:
for
x = Dx 1 +
sinh关␤共X − A兲兴
sinh共␤A兲
冎
共13兲
where x is the location of nonuniform stretched grid points, X is
the location of the uniformly distributed grid points, and A is a
constant given by 关35,36兴
A=
冋
1 + 共e␤ − 1兲共Dx/L兲
1
ln
2␤
1 + 共e−␤ − 1兲共Dx/L兲
册
共14兲
The parameter ␤ is a stretching constant, Dx is the location of grid
clustering in the x direction, and L is the channel length. This grid
stretching is used in the x direction. However, another transformation is used in the y direction and is given as 关36兴
NOVEMBER 2007, Vol. 129 / 1519
y = DH
冦
共2Dy + ␥兲
冉 冊
冋 冉 冊
共Y−Dy兲/共1−Dy兲
␥+1
␥−1
␥+1
␥−1
共2Dy + 1兲 1 +
+ 2Dy − ␥
共Y−Dy兲/共1−Dy兲
册
冧
Table 1 Grid independence study for ␴ = 0, Re= 800, ER= 2.0:
values of xr, x1, and x2, as defined in Fig. 1
共15兲
where Dy represents the location of grid clustering in the y direction, ␥ is a stretching constant in the y direction, and Y is the
location of the uniformly distributed grid points in the y direction.
Figure 2共a兲 shows the computational nonuniform mesh used in the
present work. The algebraic finite volume equations for the momentum and energy equations, in discretized form, are written into
the following form:
− a E␾ E − a W␾ W + a P␾ P = a N␾ N + a S␾ S + b
兺 兺
␧=
兺 兺
i=N
j=1
i=1
兩residij兩
j=M
i=N
j=1
i=1
⬍ 10−5
兩␾ij兩
共17兲
where ␧ is the tolerance and “resid” is the residual; M and N are
the number of grid points in the x and the y directions, respectively.
After solving for u, v, and T, further useful quantities are obtained. For example, the Nusselt number can be expressed as:
Nu =
h共DH兲
k
xr
x1
x2
13⫻ 24
25⫻ 50
37⫻ 75
49⫻ 100
75⫻ 150
101⫻ 199
125⫻ 250
151⫻ 299
6.50
4.00
3.35
5.77
5.90
6.00
6.00
6.00
Not predicted
2.50
2.35
4.65
4.81
4.81
4.81
4.81
Not predicted
5.20
5.47
9.45
9.76
10.10
10.14
10.15
共16兲
where P, W, E, N, and S denote cell location, west face of the
control volume, east face of the control volume, north face of the
control volume, and south face of the control volume, respectively
共see Fig. 2共b兲兲 and b is a source term. The symbol ␾ in Eq. 共13兲
holds for u, v, or T. The resulting algebraic equations are solved
using the tridiagonal matrix algorithm 共Thomas algorithm兲 with
the line-by-line relaxation technique. The convergence criteria
were defined by the following expression:
j=M
Grid Size
共18兲
The heat transfer coefficient is expressed as
␶w = ␮
Using the definition of nondimensional quantities, the shear
stress at the wall is expressed as
␮um du
DH dy
␶w =
qw
Tw − Tb
Cf =
2 du
Re dy
* C f Re
Cf =
2
C favg =
共20兲
⳵ T/⳵ *y
冕冑
冏
1 ⳵␪
␪w − ␪b ⳵ y
冏
冏
冏
共22兲
y=1/2
1/2
␪b =
u␪dy
−1/2
1/2
udy
−1/2
The coefficient of friction is defined as 关29兴
Cf =
␶w
2
1/2␳um
The shear stress at the wall 共␶w兲 is expressed as
1520 / Vol. 129, NOVEMBER 2007
L
共27兲
冕冑
L
关Nu共x兲兴2dx
Nuavg =
0
L
共28兲
Grid Testing
where ␪b is bulk temperature, defined as
冕
冕
0
共21兲
y=−1/2
Similarly, the Nusselt number on the top wall is written as
1 ⳵␪
Nu = −
␪w − ␪b ⳵ y
*
共C f 共x兲兲2dx
Similarly, the average Nusselt number is defined as
By substituting Eqs. 共19兲 and 共20兲 into Eq. 共18兲, and using the
nondimensional quantities, the Nusselt number on the bottom wall
is written as
Nu = −
共26兲
The average coefficient of friction on the top or bottom wall is
defined as
共19兲
qw
共25兲
The modified coefficient of friction is defined as
The thermal conductivity is expressed as
k=−
共24兲
Substituting Eq. 共24兲 into Eq. 共23兲, the coefficient of friction is
written as
L
h=
*
du
dy*
共23兲
Extensive mesh testing was performed to guarantee a gridindependent solution. Eight different meshes were used for Re
= 800 and ER= 2.0 as shown in Table 1. The problem selected for
the grid testing is the flow over a BFS with impermeable walls.
This is a well-known benchmark problem and is accepted as a
benchmark problem by the “Benchmark Solutions to Heat Transfer Problems” organized by the K-12 Committee of the ASME for
code validation and assessment 关4,5兴. The present code was tested
for grid independence by calculating the reattachment length 共xr兲,
beginning of the top secondary bubble 共x1兲, and end of the upper
secondary bubble 共x2兲; see Fig. 1. Table 1 reports the results obtained for the grid independence study. As shown in Table 1, a
grid size of 125⫻ 250 共125 grid points in y and 250 grid points in
x兲 ensures a grid-independent solution. Moreover, a gridindependence test is carried out for the permeable case for ␴ =
−0.005 and Re= 800 as shown in Table 2. It is clear that a grid size
of 125⫻ 250 ensures a grid independent solution.
Transactions of the ASME
Table 2 Grid independence study for ␴ = −0.005, Re= 800, ER
= 2.0: values of xr, x1, and x2, as defined in Fig. 1
Grid Size
xr
x1
x2
13⫻ 24
25⫻ 50
37⫻ 75
49⫻ 100
75⫻ 150
101⫻ 199
125⫻ 250
151⫻ 299
5.24
2.39
2.47
2.87
3.60
3.73
3.75
3.75
3.10
1.47
1.58
1.93
2.34
2.60
2.61
2.62
3.97
4.24
6.30
9.11
11.10
12.05
12.17
12.17
Code Validation
The present numerical solution is validated by comparing the
present code results for the benchmark problem for Re= 800 and
ER= 2.0 to the experiment of Armaly et al. 关1兴 and to other numerical published data 关2–6兴. The problem selected for the grid
testing is the flow over a BFS with impermeable walls. As shown
in Table 3, the code results are very close to the previous numerical published results. However, all of the numerical published
works, including the present one, underestimate the reattachment
length. According to Armaly et al. 关1兴, the flow at Re= 800 has
three-dimensional features. Therefore, the underestimation of xr,
by all numerical published data is due to the two-dimensional
assumption embedded in the numerical solution 关1,4,37兴. More
specifically, it is due to the sidewall-induced three-dimensional
effects 关19,20兴. According to Williams and Baker 关19兴, the interaction of a wall jet at the step near the sidewalls with the mainstream flow causes formation of three-dimensional flow structure
in a region of essentially two-dimensional flow near the midplane
of the channel. Thus, the underestimation of the reattachment
length at high values of Reynolds number is due to the influence
of the three-dimensional effects due to sidewalls 关19,20兴. Note
that Barkely et al. 关25兴 have shown that in the absence of sidewalls the transition to three-dimensional flow structures appears at
higher values of Reynolds number of ⬃1000.
Further comparisons between present code and previously published data, for ER= 3, 2, and 3 / 2, are given in Fig. 3共a兲, which
shows good agreement between the present results and the experiment of Armaly et al. 关1兴 for Reⱕ 600. Also, it shows good agreement with all numerical published data for the whole range of
Reynolds numbers and the whole range of expansion ratios. Furthermore, comparisons of temperature and x component of velocity profiles are shown in Figs. 3共b兲–3共d兲, also showing good
agreement between the present and previous investigations.
Results and Discussion
The range of Reynolds number used in the present work is
200ⱕ Reⱕ 800. The range of bleed coefficient is taken as
−0.005ⱕ ␴ ⱕ 0.005. The Prandtl number is kept constant at 0.71.
Figure 4共a兲 shows that the reattachment length xr distribution
versus bleed coefficient. It is clear that suction decreases the value
of xr. This is due to the streamlines’ attraction to the bottom wall.
Table 3 Validation tests for Re= 800: values of xr, x1, and x2
„defined in Fig. 1… for previous investigations and present work
Authors
Type of Work
xr
x1
x2
Armaly et al. 关1兴
Vradis et al. 关5兴
Kim and Moin 关3兴
Gartling 关2兴
Pepper et al. 关6兴
Present work
Experimental
Numerical
Numerical
Numerical
Numerical
Numerical
7.20
6.13
6.00
6.10
5.88
6.00
5.30
4.95
No data
4.85
4.75
4.81
9.40
8.82
No data
10.48
9.80
10.15
Journal of Heat Transfer
Fig. 3 Model validation: „a… reattachment length versus Reynolds number for various expansion ratios; „b… temperature
profiles, Re= 800; „c… velocity profiles, x = 3 and 7, Re= 800; and
„d… velocity profiles, x = 15 and 30, Re= 800
NOVEMBER 2007, Vol. 129 / 1521
Fig. 4 „a… Reattachment length, „b… beginning of secondary
recirculation bubble on the top wall, „c… end of secondary recirculation bubble on the top wall, and „d… length of top-wall
recirculation zone
On the other hand, for the case of blowing and for the case or
Re= 800, the reattachment length first increases with blowing until
it reaches a maximum value, then it starts decreasing. The reason
for this behavior is that, for high values of bleed coefficient, blowing forces the flow to detach. Thus, there is a specific value of
blowing bleed coefficient corresponding to the maximum reattachment length. Figures 4共b兲 and 4共c兲 show the beginning and the
1522 / Vol. 129, NOVEMBER 2007
Fig. 5 Distribution of coefficient of friction: „a… Bottom wall
Re= 400 „b… bottom wall Re= 800, „c… top wall Re= 400, and „d…
top wall Re= 800
end of the upper recirculation zone, respectively. Moreover, Fig.
4共d兲 shows the length of the secondary bubble. Also, Figs.
4共b兲–4共d兲 show that the secondary bubble only exists for values of
␴ lower 共or slightly higher兲 than 0.00125.
Figures 5共a兲 and 5共b兲 present the distribution of the modified
Transactions of the ASME
Fig. 6 x component of velocity „u… and velocity gradients profiles„du / dy… for Re= 400: „a… u at x = 2.0, „b…
du / dy at x = 2.0, „c… u at x = 3.0, „d… du / dy at x = 3.0, „e… u at x = 5.50, „f… du / dy at x = 5.50, „g… u at x = 8.0, and
„h… du / dy at x = 8.0
coefficient of friction for various values of bleed coefficient on the
bottom wall for Re= 400 and 800, respectively. The modified coefficient of friction is negative inside the recirculation bubble due
to the backflow. The backflow is recognized by the negative values of velocity and negative velocity gradients as shown in Figs.
Journal of Heat Transfer
6共a兲 and 6共b兲. At the point of reattachment, the coefficient of
friction is zero due to the vanished velocity gradients.
By examining the effect of blowing on modified coefficient of
friction, within the recirculation bubble, it is clear that blowing
decreases the coefficient of friction. This is due to the repulsion of
NOVEMBER 2007, Vol. 129 / 1523
streamlines from the bottom wall. Therefore, blowing reduces velocity gradients, as shown in Figs. 6共a兲–6共d兲, and accordingly
reduces the coefficient of friction. This behavior prevails after the
point of reattachment, but a second opposing effect takes place
beyond this point, which is the mass augmentation. The mass
augmentation causes an increase in velocity gradients, which results in higher values of the coefficient of friction. This is clearly
demonstrated in Figs. 6共e兲–6共h兲 where the velocity gradients increase in the x direction. Thus, the reduction of coefficient of
friction due to streamline repulsion is counteracted by mass augmentation effect. These two opposing factors reach a point where
their net effect on modified coefficient of friction diminishes. This
point is identified as the intersection of the blowing line with the
impermeable wall case.
The effect of suction on the modified coefficient of friction is
opposite to the effect of blowing. Before the reattachment point,
suction increases the coefficient of friction due to streamlines attraction. However, after the reattachment point the mass reduction, due to suction, reduces coefficient of friction. It is important
to note that for the case of suction the coefficient of friction has a
peak value after the reattachment point. This peak coincides with
the appearance of the secondary recirculation bubble on the top
wall. The top secondary bubble narrows down the flow passage
and maximizes local velocity gradients on the bottom wall.
The distribution of the coefficient of friction on the top wall is
shown in Figs. 5共c兲 and 5共d兲 for Re= 400 and 800, respectively.
The effect of suction and blowing on the modified coefficient of
friction on the top wall is opposite to that on the bottom wall. At
the top wall, the flow develops as a laminar boundary layer where
the maximum coefficient of friction is detected at the leading edge
of the top wall, as shown in Figs. 5共c兲 and 5共d兲. The thickness of
the boundary layer on the top wall is reduced by blowing, which
results in an increase in the coefficient of friction. It is worth
mentioning that the laminar boundary layer on the top wall is
sensitive to adverse pressure gradient due to geometry expansion.
By increasing the Reynolds number, the adverse pressure gradient
increases, which leads to flow separation in the top-wall boundary
layer. The first separation point on the top wall is identified by the
beginning of the secondary recirculation bubble. The effect of
suction is to widen the secondary bubble by repulsion of streamlines from the top wall. This is demonstrated in Fig. 5共c兲. However, blowing shrinks the secondary bubble by delaying boundary
layer separation or eliminating it for a relatively high bleed coefficient, as shown in Figs. 5共c兲 and 5共d兲.
Figures 7共a兲–7共c兲 present streamline patterns for Re= 400,
where the effect of suction and blowing on the primary recirculation bubble and the secondary recirculation bubble is clearly demonstrated. Figure 8 shows the distribution of average coefficient of
friction versus bleed coefficient. On the bottom wall, the flow at
Re= 800 is more sensitive to bleed than that of Re= 400. A comprehensive view for the distribution of the coefficient of friction
on the bottom and top walls is presented in Fig. 9, where contour
maps are presented for the average modified coefficient of friction
on the bottom and top walls. Figure 9 is very useful in characterizing the spots of high values of coefficient of friction. For example, at the bottom wall, high values of the modified coefficient
of friction correspond to the high suction bleed coefficient and
high Reynolds numbers. However, low values of coefficient of
friction are identified by high values of Reynolds numbers and
high blowing bleed rates. On the top wall, the high values of the
coefficient of friction are characterized by high blowing bleed
coefficients. However, lower values correspond to high Reynolds
numbers and high suction bleed rates.
Figures 10共a兲 and 10共b兲 present the variation of Nusselt number
on the bottom wall and show that Nusselt number increases by
suction and decreases by blowing. Suction increases the velocity
gradient on the bottom wall, which leads to an increase in the
temperature gradient; see Figs. 11 and 12. By studying Fig. 4共a兲,
Figs. 10共a兲 and 10共b兲 simultaneously, it is clear that the maximum
1524 / Vol. 129, NOVEMBER 2007
Fig. 7 Streamline patterns for Re= 400: „a… ␴ = 0, „b… ␴ =
−0.005, „c… ␴ = 0.005
value of Nusselt number on the bottom wall coincides with the
point of reattachment. Besides, Fig. 10 shows that the local maximum in the Nusselt number distribution disappears for high values of the blowing bleed coefficient. The reason for this behavior
is that, for high values of the blowing bleed coefficient, the flow is
forced to detach. Figures 10共c兲 and 10共d兲 show the effect of the
bleed coefficient on the Nusselt number for the top wall. It is clear
that Nusselt number increases by blowing and decreases by suction, which is contrary to what happens on the bottom wall. The
reason is that suction at the bottom wall repulses streamlines from
the top wall. This leads to lower temperature gradients on the top
wall and accordingly lower values of Nusselt number.
Figure 13 presents the average Nusselt number on the bottom
and top walls. For the bottom wall, the average Nusselt number
increases by suction and decreases by blowing. Also, it is clear
that average Nusselt number on the bottom wall is more sensitive
to mass bleeding than on the top wall. This is because bleed is
imposed on the bottom wall. Figure 14 shows contour maps for
the average Nusselt number on the bottom and top walls. The
maximum values of the average Nusselt number, on the bottom
wall, are identified by high values of Reynolds number and high
suction bleed rates. Lower values of Nusselt number correspond
Fig. 8 Average coefficient of friction versus bleed coefficient
Transactions of the ASME
Fig. 9 Contour maps of modified coefficient of friction: „a…
Bottom wall and „b… top wall
to a high blowing bleed coefficient and high Reynolds numbers.
On the other hand, for the top wall, high values of the average
Nusselt numbers are found by high Reynolds numbers and high
blowing bleed rates. However, lower values of Nusselt numbers
are recognized by high suction bleed rates regardless of the value
of the Reynolds number. Figure 14 can be used to identify the
regions where high Nusselt numbers are detected and the regions
of low Nusselt numbers. This could be used from an engineering
viewpoint for the purpose of heat transfer enhancement.
Finally, it is interesting to comment on the effect of blowing on
the flow stability and, thus, on the validity of the steady flow
assumption. Blowing is expected to slightly affect backflow
共equivalently, the velocity ratio values of the shear-layer-type velocity profiles after the sudden expansion兲 and thus have only a
small effect on local absolute instability 共see 关38兴兲. Given that the
levels of backflow in the present problem are low 共i.e., absolute
instability is not strong兲, we expect that, in the absence of external
noise, the effect of blowing on flow stability is not significant. If,
on the other hand, the blowing stream is characterized by nonnegligible levels of sustained noise, then the flow can exhibit unsteadiness via convective instabilities 共see 关24兴兲.
Conclusions
The results of the present work show that on the bottom wall,
and inside the primary recirculation bubble, suction increases the
Journal of Heat Transfer
Fig. 10 Distribution of Nusselt number: „a… Bottom wall Re
= 400, „b… bottom wall Re= 800, „c… top wall Re= 400, and „d… top
wall Re= 800
modified coefficient of friction and blowing reduces it. The reattachment length of the primary recirculation bubble is increased
by increasing the blowing bleed coefficient and is decreased by
increasing the suction bleed rate. Suction increases the size of the
NOVEMBER 2007, Vol. 129 / 1525
Fig. 12 Temperature isocontours for Re= 400: „a… ␴ = 0, „b… ␴
= −0.005, and „c… ␴ = 0.005
a ⫽ upstream channel height 共m兲
C f ⫽ coefficient of friction, dimensionless
*
C f ⫽ modified coefficient of friction, dimensionless
f
Cavg ⫽ average coefficient of friction, dimensionless
Dx ⫽ location of grid clustering in x direction 共m兲
Dy ⫽ location of grid clustering in y direction 共m兲
DH ⫽ hydraulic diameter at inlet 共DH = 2a兲 共m兲
ER ⫽ expansion ratio 共H / S兲, dimensionless
H ⫽ downstream channel height 共m兲
h ⫽ local convection heat transfer coefficient
共W m−2 K−1兲
k ⫽ thermal conductivity 共W m−1 K−1兲
L ⫽ length of the channel 共m兲
M ⫽ number of points in horizontal direction
N ⫽ number of points in vertical direction
Nu ⫽ Nusselt number 共h共DH兲 / k兲
Fig. 11 Temperature profiles Re= 400: „a… x = 2.0, „b… x = 3.0, „c…
x = 5.50, and „d… x = 8.0
secondary bubble and blowing reduces it. The local Nusselt number on the bottom wall increases by suction and decreases by
blowing, and the opposite occurs at the top wall. The maximum
local Nusselt number on the bottom wall coincides with the point
of reattachment. High values of average Nusselt number on the
bottom wall are identified by high Reynolds numbers and high
suction bleed rates. However, the low values correspond to high
blowing rates. On the top wall, high Nusselt numbers are identified by high Reynolds numbers and high bleed rates.
Nomenclature
A ⫽ constant in the grid stretching equation
1526 / Vol. 129, NOVEMBER 2007
Fig. 13 Average Nusselt number distribution versus bleed coefficient for Re= 800 and Re= 400
Transactions of the ASME
*y ⫽ vertical distance 共m兲
Y ⫽ location of uniformly distributed grid points in
the y direction
Greek Letters
␣ ⫽
␤ ⫽
␥ ⫽
␧ ⫽
␪ ⫽
␮ ⫽
␳ ⫽
␴ ⫽
thermal diffusivity 共m2 / s兲
clustering parameter in x direction
clustering parameter in y direction
numerical tolerance
dimensionless temperature
dynamic viscosity 共Ns m−2兲
density 共kg m−3兲
* /u
bleed coefficient, ␴ = v
Subscripts
avg
b
LW
m
resid
UW
w
wc
wh
average
bulk value
bottom wall
mean value
residual
top wall
wall
cold wall
hot wall
w
m
␶ ⫽ shear stress 共N m−2兲
␾ ⫽ transport quantity
␷ ⫽ kinematic viscosity 共m2 / s兲
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
References
Fig. 14 Contour maps of average Nusselt number: „a… Bottom
wall and „b… top wall
p
*p
Pr
Re
qw
S
T
Tw
Tb
um
⫽ dimensionless pressure
pressure 共N m−2兲
Prandtl number 共␷ / ␣兲
Reynolds number 共 umDH ␷ 兲
heat flux 共W m−2兲
step height 共m兲
temperature 共K兲
wall temperature 共K兲
bulk temperature 共K兲
average velocity of the incoming flow at inlet
共m/s兲
u ⫽ dimensionless x component of velocity
*u ⫽ x component of velocity 共m/s兲
v ⫽ dimensionless y component of velocity
* ⫽ y component of velocity 共m/s兲
v
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
Ⲑ
vw ⫽ dimensionless bleed velocity
* ⫽ bleed velocity 共m/s兲
v
w
x ⫽ dimensionless horizontal coordinate
*x ⫽ horizontal distance 共m兲
x1 ⫽ beginning of the secondary recirculation
bubble, dimensionless
x2 ⫽ end of the secondary recirculation bubble,
dimensionless
xr ⫽ reattachment length, dimensionless
X ⫽ location of uniformly distributed grid points
y ⫽ dimensionless vertical coordinate
Journal of Heat Transfer
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Transactions of the ASME